1
The Monopolist's Market with Discrete Choices and Network Externality
Revisited:Small-Worlds, Phase Transition and Avalanches in an ACE Framework .
Denis Phan1, Stephane Pajot1, Jean Pierre Nadal2,
1 ENST de Bretagne, Département ESH & ICI - Université de Bretagne Occidentale, Brest1 Laboratoire de Physique Statistique, Ecole Normale Supérieure, Paris.
[email protected] - [email protected]
Ninth annual meeting of the Society of Computational Economics University of Washington, Seattle, USA, July 11 - 13, 2003
9th Society of Computational Economics, [email protected] 2
In this paper, we use Agent-based Computational
Economics
and mathematical theorising as complementary tools
Outline of this paper (first investigations)
1 - Modelling the individual choice in a social context Discrete choice with social influence: idiosyncratic and interactive
heterogeneity
2 - Local dynamics and the network structure (basic features)
Direct vs indirect adoption, chain effect and avalanche process From regular network towards small world : structure matters
3 - « Classical » issues in the « global » externality case Analytical results in the simplest case (mean field) « Classical » supply and demand curves static equilibrium
4 - Exploration of more complex dynamics at the global level « Phase transition », demand hysteresis, and Sethna’s inner
hysteresis Long range (static) monopolist’s optimal position and the network’s
structure
9th Society of Computational Economics, [email protected] 3
The demand side: I - modelling the individual choice in a social context
Discrete choice model with social influence :
(1) Idiosyncratic heterogeneity
ii i i i t i t
0,1max V H (t) S ( ) P
Agents make a discrete (binary) choice i in the set : {0, 1}
Surplus : Vi(i) = willingness to pay – price repeated buying
willingness to pay (1) Idiosyncratic heterogeneity : Hi(t) Two special cases (Anderson, de Palma, Thisse 1992) : « McFaden » (econometric) : Hi(t) = H + i for all t ; i ~ Logistic(0,)
Physicist’s quenched disorder (e.g. Random Field ) used in this paper
« Thurstone » (psychological): Hi(t) = H + i (t) for all t ; i (t) ~
Logistic(0,) Physicist’s annealed disorder (+ad. Assumptions : Markov Random
Field ) Also used by Durlauf, Blume, Brock among others…
Properties of this two cases generally differ (except in mean field for this model )
9th Society of Computational Economics, [email protected] 4
t i ik kk
S ( ) J . (t)
Myopic agents (reactive) : no expectations :
each agent observes his neighbourhood Jik measures the effect of the agent k ’s choice on
the agent i ’s willingness to pay: 0 (if k = 0 ) or Jik (if k = 1 )
Jik are non-equivocal parameters of social influence
(several possible interpretations)
The demand side: I - modelling the individual choice in a social context
Discrete choice model with social influence
(2) Interactive (social) heterogeneityWillingness to pay (2) Interactive (social) heterogeneity :
St(-i)
ik kiJ
J J J 0N
In this paper, social influence is assumed to be positive, homogeneous, symmetric and normalized across the neighbourhood)
9th Society of Computational Economics, [email protected] 5
The demand side: II - Local dynamics and the network structure 1 - Direct versus indirect adoption,chain effect and avalanche process
Indirect effect of prices: « chain » or « dominoes »
effectVariation in price
( P1 P2 )
Change of agent i
Change of agent k
k t i 1 2
k t i 2 2
H S ( P ) P
H S ( P ) P
i t i 1 2H S ( P ) P
Variation in price
( P1 P2 )
Change of agent i
Change of agent j
Direct effect of prices
An avalanche carry on as long as:
k t 1 i 2 2
k t i 2 2
k / H S ( P ) P
H S ( P ) P
9th Society of Computational Economics, [email protected] 6
The demand side: II - Local dynamics and the network structure 2 - From regular network towards small
world : structure matters (a)
Total connectivity
Regular network (lattice)
Small world 1(Watts Strogatz)
Random network
• Milgram (1967)“ six degrees of separation”
• Watts and Strogatz (1998)• Barabasi and Albert, (1999)
“ scale free ” (all connectivity) multiplicative process power law blue agent is “hub ” or “gourou ”
9th Society of Computational Economics, [email protected] 7
The demand side: II - Local dynamics and the network structure 2 - From regular network towards small
world : structure matters (b)
0
5
10
15
20
25
30
35
40
0,650,750,850,951,051,151,251,351,451,55
Price
Nu
mb
er
of
cu
sto
mers
Empty
Neighb2
Neighb4
World
Neighb2 + SW
Neighb4 + SW
World Empty
Neighb2
Neighb4
Neighb2+ SW
Neighb4 + SW
9th Society of Computational Economics, [email protected] 8
III - « Classical » issues in the « global » externality case
1 - Analytical results in the simplest case:global externality / full connectivity (main
field)
• H > 0 : only one solution• H < 0 : two solutions ; results depends on .J
P
max (P) P. 1 F P H J. (P)
Supply SideOptimal pricing by a monopolist
in situation of risk
m
1 F(z);
with : z P H J.
Demand SideIn this case, each agent observes only
the aggregate rate of adoption, Let m the marginal consumer: Vm= 0
1
1 exp( .z)
for large populations. With F logistic :
Aggregate demandmay have two fixedpoint for high low ; (here = 20)
0.2 0.4 0.6 0.8 1
0.2
0.4
0.6
0.8
1
Optimum / implicit derivation gives (inverse) supply curve :
d f (z)
dP 1 J.f (z) P
s 1
p ( ) J..(1 )
9th Society of Computational Economics, [email protected] 9
0.2 0.4 0.6 0.8 1
1
2
3
4
5
6
J = 4
J = 0
H = 0
Ps
Pd
III - « Classical » issues in the « global » externality case
2 - Inverse curve of supply and demand: comparative static
s 1p ( ) J.
.(1 )
d 1 1
p ( ) H J. . ln
0.2 0.4 0.6 0.8 1
1
2
3
4
5
6
J = 4
J = 0
H = 2 PsPd = 1(one singleFixed point)
Dashed linesJ = 0no
externality
0.2 0.4 0.6 0.8 1
0.5
1
1.5
2
2.5
3
H = 1.9
J = 4
PsPd
Low / high P0.2 0.4 0.6 0.8 1
1
2
3
4
5
6
J = 4
H = 1
J = 0
Ps
Pd
9th Society of Computational Economics, [email protected] 10
III - « Classical » issues in the « global » externality case
3 - Phase diagram & profit regime transition
Full discussion of phasediagram in the plane
.J, .h, and numerically calculated solutions are
presented in:Nadal et al., 2003
+> -
+
-
-
+> -
+
P+
P -
9th Society of Computational Economics, [email protected] 11
IV - Exploration by ACE of more complex dynamics at the global level
1 - Chain effect, avalanches and hysteresis
0
10
20
30
40
50
60
70
80
90
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33
Chronology and sizes of induced adoptions in the avalanche when decrease from
1.2408 to 1.2407
0
200
400
600
800
1000
1200
1400
1 1,1 1,2 1,3 1,4 1,5
First order transition (strong connectivity)
i i i kk
V H J . P
P = H + JP = H
Homogeneous
population: Hi = H i
0
200
400
600
800
1000
1200
1400
1 1,1 1,2 1,3 1,4 1,5
= 5 = 20
9th Society of Computational Economics, [email protected] 12
IV - Exploration by ACE of more complex dynamics at the global
level 2 - hysteresis in the demand curve :
connectivity effectprices-customers hysteresis neighbours = 2
0
200
400
600
800
1000
1200
1400
0,9 1 1,1 1,2 1,3 1,4 1,5 1,6
prices
customers
prices-customers hysteresis neighbours = 4
0
200
400
600
800
1000
1200
1400
0,9 1 1,1 1,2 1,3 1,4 1,5 1,6
prices
customers
prices-customers hysteresis neighbours = 8
0
200
400
600
800
1000
1200
1400
1 1,1 1,2 1,3 1,4 1,5 1,6
prices
customers
prices-customers hysteresis neighbours = world
0
200
400
600
800
1000
1200
1400
1 1,1 1,2 1,3 1,4 1,5
prices
customers
9th Society of Computational Economics, [email protected] 13
IV - Exploration by ACE of more complex dynamics at the global level
(3) hysteresis in the demand curve :Sethna inner hystersis
(neighbourhood = 8, H = 1, J = 0.5, = 10) - Sub trajectory : [1,18-1,29]
0
200
400
600
800
1000
1200
1400
1,1 1,15 1,2 1,25 1,3 1,35 1,4
AB
9th Society of Computational Economics, [email protected] 14
1296 Agents optimal prices
adoptors profit Adoption rate
q
no externality 0,8087 1135 917,91 87,58%
Neighbour2 1,0259 1239 1 271,17 95,60%
Neighbour 4 1,0602 1254 1 329,06 96,76%
Neighbour 4_130x2 1,0725 1250 1 340,10 96,45% 5%
Neighbour 4_260x2 1,0810 1244 1 344,66 95,99% 10%
Neighbour 4_520x2 1,0935 1243 1 358,86 95,91% 20%
Neighbour 4_1296x2 1,1017 1237 1 362,35 95,45% 50%
Neighbour 6 1,0836 1257 1 361,48 96,99%
Neighbour 6_260x2 1,0997 1252 1 376,78 96,60% 7%
Neighbour 6_520x2 1,1144 1247 1 389,05 96,22% 13%
Neighbour 6_1296x2 1,1308 1241 1 403,03 95,76% 33%
Neighbour 6_1296x4 1,1319 1240 1 403,02 95,68% 66%
Neighbour 8 1,1009 1255 1 381,89 96,84%
Neighbour 8 260 x 2 1,1169 1249 1 395,43 96,37% 5%
Neighbour 8 520 x 2 1,1306 1245 1 407,20 96,06% 10%
Neighbour 8 1296x2 1,1461 1238 1 419,28 95,52% 25%
Neighbour 8 1296x4 1,1474 1239 1 421,97 95,60% 50%
Neighbour 8 1296x6 1,1498 1238 1 423,84 95,52% 75%
world 1,1952 1224 1 462,79 94,44%
IV - Exploration by ACE of more complex dynamics at the global level
Optimal long run (static) pricing by a monopolist: the influence of local network
structure
• optimal static (long run) monopoly prices increase with connectivity and small world parameter q ; higher with scale free than WS.
0.5 1 1.5 2
0.5
1
1.5
2
J=0
Global externality
9th Society of Computational Economics, [email protected] 15
Conclusion, extensions & future developments
Even with simplest assumptions (myopic customers, full connectivity, risky situation), complex dynamics may arise.
Actual extensions: long term equilibrium for scale free small world, and dynamic regimes with H<0.
In the future: looking for cognitive agents ….
Dynamic pricing & monopolist’s Bayesian learning process in the case of repeated buying
Dynamic pricing & agent’s learning process in the case of durable good (Coase conjecture)
Dynamic network and monopolist’s learning about the network ….
9th Society of Computational Economics, [email protected] 16
References Anderson S.P., DePalma A, Thisse J.-F.
(1992) Discrete Choice Theory of Product Differentiation, MIT Press, Cambridge MA.
Brock Durlauf (2001) “Interaction based models” in Heckman Leamer eds. Handbook of econometrics Vol 5 Elsevier, Amsterdam
Phan D. (2003) “From Agent-based Computational Economics towards Cognitive Economics”, in Bourgine, Nadal (eds.), Towards a Cognitive Economy, Springer Verlag, Forthcoming.www-eco.enst-bretagne.fr/~phan/moduleco
Phan D. Gordon M.B. Nadal J.P. (2003) “Social interactions in economic theory: a statistical mechanics insight”, in Bourgine, Nadal (eds.), Towards a Cognitive Economy, Springer Verlag, Forthcoming.
Nadal J.P. Phan D. Gordon M.B. Vannimenus J. (2003), "Monopoly Market with Externality: an Analysis with Statistical Physics and ACE", 8th Annual Workshop on Economics with Heterogeneous Interacting Agents, Kiel.
Any Questions ? (please speak slowly)